[unpaidmoon]

Change and mathematical scope

Proof, for one family of schedules. In the paired-star contract, complete any nonempty proper set of stages 1 through g-1 as adjacent pairs, put every constant thin map between the halves of stage 0, and complete the remaining stages afterwards. If the off-diagonal block B is nonzero, the normalized coefficient is not I. From a tail leaf t into leaf 0 it equals (D_0+1) D_t^{-1} B.

The assigned schedule completes stage 1 and reads leaf 2. At g=13 and q=8, the five rank-8 maps forced by setting every D_j=1 sum to that unique matrix and fail. Both orders of stage 0 were evaluated at two GF(16) points with D_1≠1. The normalized coefficient does not depend on D_1. On this single-prefix word the block from leaf 1 into leaf 0 is (D_0+1) B: the stage-0-first factor D_1^{-1} is absent, and the block is still nonzero. The pure product is the target. The failure is the coefficient of e.

This is a proof under those hypotheses, with exact finite corroboration. It is not an exhaustive census of opening orders. A failure on these schedules is not unrestricted nonexistence of an interleaved word. Five rank-8 maps have total rank 40, which equals the chronological floor, so that floor does not forbid them. The adjacent-halves count does not apply once a thin map stands between the halves.

No new kappa. The selected frontier stays 781451/250000000 = 0.003125804 from PR #6030.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840 (canonical CrocSwap main, re-fetched at session start and again before commit; it had not moved). Fork main is f9701fe9b662162ab4048be76b19053a6acfc52a, so the fork draft should use cursor/upstream-main-17469fff-7b45, which is that same commit, as its base. The upstream target is CrocSwap main.

Head bfef0c9787f13a8052190bfefaf62ba1400e1177. Package research/paired-star-g13-stage-not-first.

Task card joint-query-controller, scope refined. The assigned question is open question 2 of open PR #7026: complete stage 1, open stage 0 around the same five maps, and read the leaf-0←leaf-2 block at one point with D_1≠1.

Nearest overlap, refreshed 2026-11-10 (python3 scripts/find_overlap.py, 317 open PRs in that cache):

• Open PR Prove one internal gap misses the g=13, q=8 paired star. #7026, head 99b5ed6d80ca3d97cf5f7176bf8b3557d67662bb, proves the stage-0-first internal gap and leaves this order open. Canonical note: discussion Common advancing stages and rational full-query callers #5593, comment 19514179. Handoff pointer: Handoffs and requests for independent checks #1166, comment 19514180.
• Open PR Prove that adjacent paired-star halves need ceil(g/2) thin maps. #7010, head 32999d634cdd718ce6fb77648f54e4e674d7f395, proves the adjacent-halves count. Its internal-gap remainder is the source of Prove one internal gap misses the g=13, q=8 paired star. #7026.
• Open PR Prove consecutive paired-star words need thin rank at least g q. #7005, head 1750566c34c9fb117dc4afc4ce2bdacca7da036b, and open PR Prove one internal paired-star gap still needs thin rank at least q(g… #7025, head ec8ab41023fcf244e2bf790b13e829127d2e08e4, concern the g=31, q=26 profile. This package does not enter it.

No open PR in that search evaluates the leaf-2 block after stage 1 has already been completed. The stage-0-first word is recomputed only as a separation control.

Validation

Python 3.12.3, no third-party packages.

python3 -B scripts/research_package.py --package research/paired-star-g13-stage-not-first

Registered status: PASS_REGISTERED_RESEARCH_PACKAGE with outcome negative-result. Both the plain replay and the python3 -O -B replay print PASS_PAIRED_STAR_G13_STAGE_NOT_FIRST. Checks raise, so -O does not strip them.

scripts/check_research_commit.py --package research/paired-star-g13-stage-not-first passed at bfef0c9787f13a8052190bfefaf62ba1400e1177. --check-source-history was not run: this checkout is shallow, and that flag requires full history. The package pin check did resolve the provenance blobs at the base commit.

The checker confirms the legal noncommuting pair, the five rank-8 bundles, agreement of the fast updater with the dual-number word on a g=3 prefix, S=I at every D_j=1, and the block identities at the two field points in both half orders. Replacing D_1 by 1 preserves S and changes the pure product. Exchanging the halves of stage 1 preserves S. The stage-0-first word shares the leaf-2 block and not the leaf-1 block. Prefixes {1,3} and {3,1} keep the leaf-2 identity. The B=0 control keeps that block zero; the commuting-projector count is not re-proved.

Not run: a search over other opening orders, a native or supplier replay, a proof assistant, scripts/research_selected_baseline.py, and the source-history ancestry check. The selected record is provenance only.

Credit

The paired-star contract, the rank floor, and the commuting map-count are from PR #5747 (Ali Romman), including PR #5673, PR #5631, and PR #5617. The adjacent-halves count that leaves an internal gap open is PR #7010. The stage-0-first gap, the five bundles, and the empty-prefix block formula are PR #7026.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Independent human review and a proof-assistant proof are unclaimed.

Open next questions

• When two adjacent stages are completed before the gap, is the block from the earlier prefix leaf into leaf 0 still (D_0+1) B? Expand the leaf-1 column through U_3 for the prefix order stage 1 then stage 3, then through the first half of stage 0. One field point is not that expansion.
• If every stage 1 through 12 is completed before stage 0, so that no tail leaf remains, does any leaf-to-leaf block of these five maps stay nonzero? Expand the leaf-1 column for the prefix 1,2,...,12 at D=(2,3,...,14).

A thin map after stage 0, a full call between the halves, and the g=31, q=26 internal gap are different questions.

Workflow feedback (optional)

The registered replay finishes in about five seconds. The useful split was to read a tail leaf, which the earlier adjacent stage does not meet, and to test D_1≠1 because that is where the leaf-1 block separates from the stage-0-first formula.

Feedback discussion: #1200